Property implying weak differentiability

Question

What property does imply that a function $f \in L^1_{loc}(\Omega)$ ($\Omega \subset \mathbb{R}^n$) is weakly differentiable, namely there exists $g \in L^1_{loc}(\Omega)$ such that $\int_{\Omega} f\partial_i \phi = - \int_\Omega g\phi$, for all $\phi \in C^\infty_c(\Omega)$ and all $i=1,\ldots,n$? Clearly, if $f \in C^1_c(\Omega)$, then it is weakly differentiable, but is there a weaker assumption that will still imply it? Thank you.